# Different versions of the Compound Interest Formula ```Different versions of the Compound Interest Formula.
The general formula is:
T = P(1 + I/n)Yn
where:
T = Total amount of money
P = Principle amount of money
I = Interest as a decimal.
n = How often interest is calculated in a year.
Y= Number of years.
This formula can be used to calculate the total amount of money a single investment will
be worth if values for P, I, n and Y are given.
What if we wanted to find P instead? There are several practice problems that ask you to
find P. The best way to tackle these problems would be to solve T = P(1 + I/n)Yn for P
and use that formula for those problems.
Here is how we get that formula.
I

T  P1  
 n
Yn
We want P alone and what we have is P times another number
so divide.
Yn
T
I

1  
 n
So, P 
Yn
I

P1  
n
 
Yn
I

1



 n
T
I

1  
 n
Yn
We can use this formula to find P if we are given T, I, n and Y.
We can also use this formula to find I. This is a little tougher and it involves a technique
that hasn’t been fully explored in this course.
I

T  P1  
 n
Yn
We want to get I alone but it is inside the group that has the exponent Yn. We can move
the P by dividing.
Yn
I

P 1  
Yn
T
T 
I
n

We now have  1  

P  n
P
P
We need to do something about the exponent. Let’s use a simpler equation to
demonstrate the method to be used.
Find the exponent that makes the following equation a true statement. (34)x = 3
First, let’s review one of the rules of exponents that was introduced in Module 3.
Multiply the exponents: (34)5 = 345 = 320
Therefore, (34)x = 34x and 34x = 3 = 31
To find x we can solve the equation: 4x = 1
Thus x = &frac14; and (34)1/4 = 3.
Yn
T 
I
 1   . To get the variable I
P  n
out of the group we need the exponent to be 1. This can be done be the exponent 1/Yn to
both sides of the equation.
How do we use this? Well, we have the equation
1
Yn
1
1
Yn

I   Yn
I
T
 T  Yn

    1    which simplifies to    1  . We’re now two steps from
n
P
P
 n  
the formula we’re looking for.
1
I
 T  Yn
  1  11
n
P
1
I
 T  Yn
  1 
n
P
1


  T  Yn
 I
n     1   n
 P 
 n


1


  T  Yn

Thus , I  n     1
 P 



We can use this formula to find I if we are given T, P, n and Y.
The formula to find Y is developed in Unit 4 of this Module 8.
Here are all the formulas that are related to calculating Compound Interest.

T  P1 

I

n
Yn
is used when P, I, n and Y are given.
1


  T  Yn

I  n    1 is used when T, P, Y and n are given.
 P 



P
T
I

1  
 n
Yn
is used when T, I, n and Y are given.
T
log  
P
Y
is used when T, P, I and n are given.
 
I 
n log 1  
  n 
These formulas are used to solve the practice problems for Units 2 and 4.
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